Let T be a bounded linear operator on a (real or complex) Banach space X. If (aₙ) is a sequence of non-negative numbers tending to 0, then the set of x ∈ X such that ||Tⁿx|| ≥ aₙ||Tⁿ|| for infinitely many n’s has a complement which is both σ-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents q > 0 such that for every non-nilpotent operator T, there exists x ∈ X such that , using techniques which involve the modulus of asymptotic uniform smoothness of X.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-1-2,
author = {Jean-Matthieu Aug\'e},
title = {Orbits of linear operators and Banach space geometry},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {21-39},
zbl = {1266.47001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-1-2}
}
Jean-Matthieu Augé. Orbits of linear operators and Banach space geometry. Studia Mathematica, Tome 209 (2012) pp. 21-39. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-1-2/